Questions: Consider the following data: 5,6,11,9,7,10 Step 1 of 3: Calculate the value of the sample variance. Round your answer to one decimal place.

Solution Steps

To find the mean \( \mu \) of the dataset, we use the formula: \[ \mu = \frac{\sum x_i}{n} \] where \( \sum x_i = 5 + 6 + 11 + 9 + 7 + 10 = 48 \) and \( n = 6 \). Thus, \[ \mu = \frac{48}{6} = 8.0 \]

The sample variance \( \sigma^2 \) is calculated using the formula: \[ \sigma^2 = \frac{\sum (x_i - \mu)^2}{n-1} \] Calculating \( (x_i - \mu)^2 \) for each \( x_i \):

• For \( x_1 = 5 \): \( (5 - 8)^2 = 9 \)
• For \( x_2 = 6 \): \( (6 - 8)^2 = 4 \)
• For \( x_3 = 11 \): \( (11 - 8)^2 = 9 \)
• For \( x_4 = 9 \): \( (9 - 8)^2 = 1 \)
• For \( x_5 = 7 \): \( (7 - 8)^2 = 1 \)
• For \( x_6 = 10 \): \( (10 - 8)^2 = 4 \)

Now, summing these values: \[ \sum (x_i - \mu)^2 = 9 + 4 + 9 + 1 + 1 + 4 = 28 \] Thus, the sample variance is: \[ \sigma^2 = \frac{28}{6-1} = \frac{28}{5} = 5.6 \]

The standard deviation \( \sigma \) is the square root of the variance: \[ \sigma = \sqrt{5.6} \approx 2.4 \]

Final Answer